The optimal competition resolution rule for a controlled binary chain

A. G. Tatashev   ; M. V. Yashina

Vladikavkazskij matematičeskij žurnal, Tome 26 (2024) no. 1, pp. 142-153

Résumé

A dynamical system that belongs to the class introduced by A. P. Buslaev is investigated. The system contains $N$ contours. There are two cells and one particle on each contour. For each contour there is one common point, called a node, with each of the neighboring nodes. In the deterministic version of the system, at any discrete moment, each particle moves to another cell if there is no delay. The delays are due to the fact that two particles cannot pass through the node at the same time. If two particles tend to cross the same node, then only one particle moves in accordance with a given rule of competition resolution. In the stochastic version the particle tends to move in a state corresponding to the state of the deterministic system in which the particle is moving. This attempt is implemented in the corresponding system with a probability of $1-\varepsilon,$ where $\varepsilon$ — is a small value. A rule for resolving competition, called the long cluster rule, is obtained, such that this rule puts the system in such a state that all particles move without delay at the present moment and in the future (the state of free movement), and the system gets into a state of motion in the shortest possible time. The average number of $v_i$ displacements of a particle of the $i$-th contour per unit of time is called the average velocity of this particle, $i=1,\dots,N.$ For the stochastic version of the system, the following is established under the assumption that $N=3.$ For the long rule, the average particle velocities $v_1=v_2=v_3=1-2\varepsilon+o(\varepsilon)$ $(\varepsilon\to 0).$ For the left-priority rule, according to which, in competition, the particle of the contour with the lower number has priority, the average particle velocity $v_1=v_2=v_3=\frac{6}{7}+o(\sqrt{\varepsilon}).$

Bibliographie

[1] Schreckenberg M., Shadshneider M., Nagel K., Ito N., “Discrete stochastic models for traffic flow”, Phys. Rev., 51:4 (1995), 2939–2949 | DOI

[2] Blank, M. L., “Exact Analysis of Dynamical Systems Arising in Models of Traffic Flow”, Russian Mathematical Surveys, 55:3 (2000), 562–563 | DOI | DOI | MR | Zbl

[3] Belitsky V., Ferrari P. A., “Invariant measures and convergence properties for cellular automation 184 and related processes”, J. Stat. Phys., 118:3/4 (2005), 589–523 | DOI | MR

[4] Gray L., Griffeath D., “The ergodic theory of traffic jams”, J. Stat. Phys., 105:3/4 (2001), 413–452 | DOI | MR | Zbl

[5] Kanai M., Nishinary K., Tokihiro T., “Exact solution and asymptotic behaviour of the asymmetric simple exclusion process on a ring”, J. Phys. A: Math. Gen., 39:29 (2006), 9071 | DOI | MR | Zbl

[6] Kanai M., “Two-lane traffic-flow model with an exact steady-state solution”, Phys. Rev. E, 82 (2010), 066107 | DOI | MR

[7] Yashina M. V., Tatashev A. G., “Traffic model based on synchronous and asynchronous exclusion processes”, Math. Method. Appl. Sci., 43:14 (2020), 8136–8146 | DOI | MR | Zbl

[8] Wolfram S., “Statistical mechanics of cellular automata”, Rev. Mod. Phys., 55:3 (1983), 601–644 | DOI | MR | Zbl

[9] Spitzer F., “Interaction of Markov processes”, Advances in Math., 5:2 (1970), 246–290 | DOI | MR | Zbl

[10] Biham O., Middleton A. A., Levine D., “Self-organization and a dynamical transition in traffic-flow models”, Phys. Rev. A, 46:10 (1992), R6124–R6127 | DOI | MR

[11] Angel O., Horloyd A. E., Martin J. B., “The Jammed phase of the Biham–Middelton–Levine traffic flow model”, Electron. Commun. Probab., 10 (2005), 167–178 | DOI | MR | Zbl

[12] Moradi H. R., Zardadi A., Heydarbeygi Z., “The number of collisions in Biham–Middleton–Levine model on a square lattice with limited number of cars”, Applied Math. E-Notes, 2019, 243–249 | MR | Zbl

[13] Buslaev A. P., Yashina M. V., “On holonomic mathematical $F$-pendulum”, Math. Method. Appl. Sci., 39:16 (2016), 4820–4828 | DOI | MR | Zbl

[14] Bugaev A. S., Buslaev A. P., Kozlov V. V., Yashina M. V., “Distributed problems of monitoring and modern approaches to traffic modeling”, 14th International IEEE Conference on Intelligent Transportation Systems (ITSC), IEEE, Washington, 2011, 477–481 | DOI | MR

[15] Kozlov V. V., Buslaev A. P., Tatashev A. G., “On synergy of totally connected flows on chainmails”, Proceed. of International Conference of CMMSE (Spain, 2013), v. 3, 861–874

[16] Kozlov V. V., Buslaev A. P., Tatashev A. G., “Monotonic walks on a necklace and a coloured dynamic vector”, International Journal of Computer Math., 92:9 (2015), 1910–1920 | DOI | MR | Zbl

[17] Kozlov V. V., Buslaev A. P., Tatashev A. G., “A dynamic communication system on a network”, Journal of Computational and Applied Math., 2015, 247–261 | DOI | MR | Zbl

[18] Tatashev A. G., Yashina M. V., “Spectrum of elementary cellular automata and closed chains of contours”, Machines, 7:2 (2019), 28 pp. | DOI

[19] Buslaev A. P., Fomina M. Yu., Tatashev A. G., Yashina M. V., “On discrete flow networks model spectra: statements, simulation, hypotheses”, Journal of Physics: Conference Series, 1053 (2018), 012034, 1–7 | DOI

[20] Bugaev, A. S., Tatashev, A. G. and Yashina, M. V., “Spectrum of a Continuous Closed Symmetric Chain with an Arbitrary Number of Contours”, Mathematical Models and Computer Simulations, 13:6 (2021), 1014–1027 | DOI | DOI | MR | Zbl

[21] Yashina M. V., Tatashev A. G., “Spectral cycles and average velocity of clusters in discrete two-contours system with two nodes”, Math. Method. Appl. Sci., 43:7 (2020), 4303–4316 | DOI | MR | Zbl

[22] Yashina M. V., Tatashev A. G., Fomina M. Y., “Optimization of velocity mode in buslaev two-contour networks via competition resolution rules”, International Journal of Interactive Mobile Technologies, 14:10 (2020), 61–73 | DOI

[23] Myshkis, P. A., Tatashev, A. G. and Yashina, M. V., “Cluster Motion in a Two-Contour System with Priority Rule for Conflict Resolution”, Journal of Computer and Systems Sciences International, 59:3 (2020), 311–321 | DOI | DOI | MR | Zbl

[24] Golovneva, E. V., “On Ergodic Properties Homogeneous Markov Chains”, Vladikavkaz Math. J., 14:1 (2012), 37–46 (in Russian) | DOI | MR | Zbl

[25] Kemeny J. G., Snell J. L., Finite Markov Chains, Springer-Verlag, N. Y.–Berlin–Heidelberg–Tokyo, 1976 | MR | Zbl